Can somebody give me an example of a continuous function $F:\mathbb{R}^2\rightarrow \mathbb{R}$. Such that $$\frac{\partial F(x_1,x_2)}{\partial x_i} \geq 0 \;\; \forall x_i, \;i=1,2$$ but, $$\frac{\partial F(x_1,x_2)}{\partial x_1 \partial x_2} < 0$$
How about $F(x_1,x_2)=-e^{-x_1-x_2}$?